Nonprobabilistic Reliable LQR Design Method for Active Vibration Control of Structures with Uncertainties

The linear quadratic regulator (LQR), which allows a natural consideration of the constraints on state variables and control effort, is widely used in modern control theory for the active control of structural vibration. However, uncertainties are unavoidable in the design of active control systems. Thus, in the design process of an LQR controller, it is important to deal with these uncertainties, which are always bounded and have a great influence on the final performance of the closed-loop system. In this paper, the nonprobabilistic reliability concept is extended to the selection of weighting matrices for determining the LQR controller. First, a new uncertainty propagation analysis method for an active control system is presented. To adapt to the special problem of controller design, a novel nonprobabilistic reliability index is proposed. The reliability and safety margin of the closed-loop system can be continuously characterized by this modified nonprobabilistic reliability index. The weighing matrices used to design the LQR controller via the Riccati equation are finally determined by solving a reliability-based optimization problem. The proposed method preserves the excellent characteristics of LQR control theory and provides a quantitative method for selecting the weight matrices for LQR controller design in consideration of the inherent uncertainties. Two numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed method. The results from these examples reveal that the uncertainties in active control system design can be addressed from a reliability perspective.