“Convexifying” Linear Matrix Inequality Methods for Integrating Structure and Control Design

This paper presents a methodology in the linear matrix inequality (LMI) framework to jointly optimize the linear control law and the linear parameters in the structure. The method allows the mass matrix to contain free parameters, while employing LMI methods. The paper solves a structure design problem which bounds the covariance of selected outputs, such as interstory drifts and their velocities, in the presence of random excitations. In fact, the method simultaneously designs the structure and the controller, yielding a hybrid control. The proposed method also allows one to guarantee bounds on the peak response in the presence of bounded energy excitations. With minor modifications, the method can also guarantee bounds on the $H_{\infty }$ performance and many other convex performance criteria. The nonconvex problem is approximated by a convex one by adding a certain function to make the constraint convex. This “convexifying” function is updated with each iteration until the added convexifying function disappears at a saddle point of the nonconvex problem. This is a new contribution to both control theory and structure design.