Non-probabilistic reliability method and reliability-based optimal LQR design for vibration control of structures with uncertain-but-bounded parameters

Uncertainty is inherent and unavoidable in almost all engineering systems. It is of essential significance to deal with uncertainties by means of reliability approach and to achieve a reasonable balance between reliability against uncertainties and system performance in the control design of uncertain systems. Nevertheless, reliability methods which can be used directly for analysis and synthesis of active control of structures in the presence of uncertainties remain to be developed, especially in non-probabilistic uncertainty situations. In the present paper, the issue of vibration control of uncertain structures using linear quadratic regulator (LQR) approach is studied from the viewpoint of reliability. An efficient non-probabilistic robust reliability method for LQR-based static output feedback robust control of uncertain structures is presented by treating bounded uncertain parameters as interval variables. The optimal vibration controller design for uncertain structures is carried out by solving a robust reliability-based optimization problem with the objective to minimize the quadratic performance index. The controller obtained may possess optimum performance under the condition that the controlled structure is robustly reliable with respect to admissible uncertainties. The proposed method provides an essential basis for achieving a balance between robustness and performance in controller design of uncertain structures. The presented formulations are in the framework of linear matrix inequality and can be carried out conveniently. Two numerical examples are provided to illustrate the effectiveness and feasibility of the present method.