Methods to Compute Probabilistic Measures of Robustness for Structural Systems

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Abstract
Model uncertainty, if ignored, can seriously degrade the performance of an otherwise well-designed control system. If the level of this uncertainty is extreme, the system may even be driven to instability. In the context of structural control, performance degradation and instability imply excessive vibration or even structural failure. Robust control has typically been applied to the issue of model uncertainty through worst- case analyses. These traditional methods include the use of the structured singular value (μ-analysis), as applied to the small gain condition, to provide estimates of controller robustness. However, this emphasis on the worst-case scenario has not allowed a probabilistic understanding of robust control. Because of this, an attempt to view controller robustness as a probability measure is presented. As a result, a much more intuitive insight into controller robustness can be obtained. In this context, the joint probability distribution is of dimension equal to the number of uncertain parameters, and the failure hypersurface is defined by the onset of instability of the closed-loop system in the eigenspace. A first-order reliability measure (FORM) of the system is computed and used to estimate controller robustness. It is demonstrated via an example that this FORM method can provide accurate results on the probability of failure despite the potential complexity of the closed-loop. In addition to the FORM method, a probabilistic measure of robustness is developed based on the fundamentals of μ-analysis. It is shown that the μ-analysis based method is inferior to the FORM method and can only have qualitative value when assessing control system robustness in a probabilistic framework.