Efficient Integration of the Time Varying Closed-Loop Optimal Control Problem

Much of the research done in recent years towards the development of the smart or adaptive structure focuses on the application of active control to alleviate undesirable structural responses. However, classical optimal control algorithms are not directly applicable to most structural engineering applications because the control gains, obtained by solving the matrix differential Riccati equation (DRE), neglect the effects of the external forcing function and assume a time invariant system (which allows the DRE to be reduced to an algebraic Riccati equation). Due to these simplifications, the control algorithm loses its optimality and consequently may induce significant control inefficiency. In this study, a matrix-valued integration procedure is formulated for and applied to the differential Riccati equation with time variant plant and weighting matrices. The ability to effectively integrate the time variant differential Riccati equation allows the control algorithm to adapt to changing structural parameters. Examples are presented which apply this new procedure to solve the control equations associated with a five story, tendon-controlled shear building with time variant stiffness and damping matrices.