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Dual forms and a proposed forward integration method for the matrix differential Riccati equation

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. Classical optimal control algorithms are not directly applicable to most civil engineering applications because the control gains neglect the effects of the external forcing function and assume a time invariant system (which allows the differential Riccati equation (DRE) to be reduced to an algebraic Riccati equation). The reason for these assumptions is that the time dependent DRE can only be stably integrated backwards in time. This study presents a more effective LQR control algorithm for civil structure applications, based on a proposed methodology for forward integrating the DRE. A set of dual equations are presented together with an optimization technique for obtaining the DRE solution from a forward integrable dual form. In addition, a matrix-valued integration procedure is formulated for and applied to the differential Riccati equation with time variant plant and weighting matrices.