Automatic determination of LQR weighting matrices for active structural control

This paper presents a method for the automatic selection of weighting matrices for a linear-quadratic regulator (LQR) in order to design an optimal active structural control system. The weighting matrices of a control performance index, which are used to design optimal state-feedback gains, are usually determined by rule of thumb or exhaustive search approaches. To explore an easy way to select optimal parameters, this paper presents a method based on Bayesian optimization (BO). A 10-degree-of-freedom (DOF) shear building model that has passive-base isolation (PBI) under the building is used as an example to explain the method. A control performance index that contains the absolute acceleration, along with the inter-story drift and velocity of each story, is chosen for the design of the controller. An objective function that contains the maximum absolute acceleration of the building is chosen for BO to produce optimal weighting matrices. In the numerical example, a restriction on the displacement of the PBI is used as a constraint for the selection of weighting matrices. First, the BO method is compared to the exhaustive search method using two parameters in the weighting matrices to illustrate the validity of the BO method. Then, thirty-three parameters (which are automatically optimized by the BO method) in the weighting matrices are used to elaborately tune the controller. The control results are compared to those for the exhaustive search method and conventional optimal control, in terms of the control performance of the relative displacement, absolute acceleration, inter-story-drift angle, and the story-shear coefficient of each story. The damping ratio for each mode, and the control energy and power are also compared. The comparison demonstrates the validity of the method.