Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT
Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT
3.1 Uncertainty Historically, structural models are used along with test data to help engineers better understand the physical behavior of the hardware. Model predictions are compared to test data, and the data is used to provide deeper understanding of the underlying physics and validate the model for use in trade analyses. However, as space systems become more complex and more difficult to test on the ground, test data is harder to obtain and it is necessary to rely solely on models and simulations to provide predictions of future system behavior without the benefit of validation data. This task is a much more demanding one and requires a high level of confidence in the models. As a result, much attention is focused in the area of prediction accuracy and model uncertainty. The sources of inaccuracy in model predictions are grouped into three main cate- gories: parametric errors, discretization errors and model structure errors [27, 93, 12]. Parametric errors refer to inaccuracies in the values of model parameters. For exam- ple, a finite element model may consist of beam elements that have certain material properties, such as Young’s Modulus. Although the value of Young’s Modulus is published for a wide range of materials it is likely that components made from the same material have slightly different Young’s Modulus values. A value of Young’s modulus in a finite element model that is slightly different from the Young’s modulus of the physical component is an example of a parametric error. Discretization errors exist because finite element models are composed of discrete elements while physical parts are continuous. These errors can be reduced by using a high-fidelity mesh, but not eliminated. Finally, model structure errors are global modeling omissions or mistakes. This category includes any physical system behavior that is not captured in the model. Examples of these types of errors are unmodelled nonlinearities and improper choice of element types. Parametric errors are most often considered in stochastic analysis because they are the easiest to model and the hardest to reduce. Discretization error can be reduced by refining the finite element model mesh, and model structure errors are reduced 72
through careful and experienced modeling [57]. Parametric errors, on the other hand, are nearly impossible to eliminate entirely since the model is built and used to make predictions well before components are available for testing. Even when test data is available, measurements are affected by noise in the sensor and data acquisition making it nearly impossible to obtain accurate data for model updating. Therefore, there is always some level of uncertainty inherent in the parameter values. 3.1.1 Uncertainty Models The modeling of parametric uncertainties is currently a popular field of research as engineers move away from purely deterministic models and analysis to stochastic analogues. There are three accepted ways of modeling parametric uncertainty: prob- abilistic models, fuzzy logic, and convex, i.e. bounded, models [41]. In probabilistic modeling, a random distribution, usually normal, is assigned to a parameter and the uncertainty model is defined by choosing the mean parameter value and a standard deviation. These models are propagated through the analysis to provide statistical information about the performance such as probability of mission success [19, 52]. One drawback to probabilistic modeling is that there are generally insufficient data to accurately determine the statistical properties. In effect, the uncertainty model is itself uncertain. Convex, or bounded, uncertainty models address this concern by taking a more conservative approach. The uncertain parameter is assumed to be distributed between some bounds, and the worst-case performance prediction is con- sidered. The uncertainty model used in this thesis is a type of convex model known as “envelope bounds.” For a complete treatment of convex uncertainty models see the monograph by Elishakoff and Ben-Haim [15]. 3.1.2 Uncertainty Analysis Once a model of the parametric uncertainty is chosen, the effects of the uncertainty on the model predictions are assessed through an uncertainty analysis. Quite a few techniques for propagating uncertainty through a structural model are found in the 73
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through careful and experienced modeling [57]. Parametric errors, on the other hand,<br />
are nearly impossible to eliminate entirely since the model is built and used to make<br />
predictions well before components are available for testing. Even when test data<br />
is available, measurements are affected by noise in the sensor and data acquisition<br />
making it nearly impossible to obtain accurate data for model updating. Therefore,<br />
there is always some level of uncertainty inherent in the parameter values.<br />
3.1.1 Uncertainty Models<br />
The modeling of parametric uncertainties is currently a popular field of research as<br />
engineers move away from purely deterministic models and analysis to stochastic<br />
analogues. There are three accepted ways of modeling parametric uncertainty: prob-<br />
abilistic models, fuzzy logic, and convex, i.e. bounded, models [41]. In probabilistic<br />
modeling, a random distribution, usually normal, is assigned to a parameter and the<br />
uncertainty model is defined by choosing the mean parameter value and a standard<br />
deviation. These models are propagated through the analysis to provide statistical<br />
information about the performance such as probability of mission success [19, 52].<br />
One drawback to probabilistic modeling is that there are generally insufficient data<br />
to accurately determine the statistical properties. In effect, the uncertainty model<br />
is itself uncertain. Convex, or bounded, uncertainty models address this concern by<br />
taking a more conservative approach. The uncertain parameter is assumed to be<br />
distributed between some bounds, and the worst-case performance prediction is con-<br />
sidered. The uncertainty model used in this thesis is a type of convex model known<br />
as “envelope bounds.” For a complete treatment of convex uncertainty models see<br />
the monograph by Elishakoff and Ben-Haim [15].<br />
3.1.2 Uncertainty Analysis<br />
Once a model of the parametric uncertainty is chosen, the effects of the uncertainty<br />
on the model predictions are assessed through an uncertainty analysis. Quite a few<br />
techniques for propagating uncertainty through a structural model are found in the<br />
73
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