Robust Control of Stochastic Structures Using Minimum Norm Quadratic Partial Eigenvalue Assignment Technique

The use of active vibration control technique, though typically more expensive and difficult to implement compared to the traditionally used passive control approaches, is an effective way to control or suppress dangerous vibrations in structures caused by resonance or flutter due to the external disturbances, such as winds, waves, moving weights of human bodies, and earthquakes. From control perspective, it is not only sufficient to suppress the resonant modes of vibration but also to guarantee the overall stability of the structural system. In addition, the no-spillover property of the remaining large number of frequencies and the corresponding mode shapes must be maintained. To this end, the state-space based linear quadratic regulator (LQR) and \(H_{\infty }\) control techniques and physical-space based robust and minimum norm quadratic partial eigenvalue assignment (RQPEVA and MNQPEVA) techniques have been developed in the recent years. In contrast with the LQR and \(H_{\infty }\) control techniques, the RQPEVA and MNQPEVA techniques work exclusively on the second-order model obtained by applying the finite element method on the vibrating structure, and therefore can computationally exploit the nice properties, such as symmetry, sparsity, bandness, positive definiteness, etc., inherited by the system matrices associated with the second-order model. Furthermore, the RQPEVA and MNQPEVA techniques guarantee the no-spillover property using a solid mathematical theory. The MNQPEVA technique was originally developed for the deterministic models only, but since then, further extension to this technique has been made to rigorously account for parametric uncertainty. The stochastic MNQPEVA technique is capable of efficiently estimating the probability of failure with high accuracy to ensure the desired resilience level of the designed system. In this work, first a brief review of the LQR, \(H_{\infty }\), and MNQPEVA techniques is presented emphasizing on their relative computational advantages and drawbacks, followed by a comparative study with respect to computational efficiency, effectiveness, and applicability of the three control techniques on practical structures. Finally, the methodology of the stochastic MNQPEVA technique is presented and illustrated in this work using a numerical example with a representative real-life structure and recorded data from a California earthquake.