Probabilistic Stability Robustness of Structural Systems

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, 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. In this paper an attempt to view controller robustness as a probability measure is presented. The probability of failure due to parametric uncertainty is estimated using first-order reliability methods (FORM). It is demonstrated that FORM can provide quite accurate results on the probability of failure of actively controlled structures. Moreover, a comparison of FORM to a suitably modified structured singular value robustness analysis in a probabilistic framework is performed. It is shown that FORM is the superior analysis technique when applied to the class of problems discussed. In addition, the robustness qualities of various active control design schemes such as the linear quadratic regulator (LQR), H₂, H_∞, and μ-synthesis are discussed in order to provide some design guidelines.