Closed-Form Solutions to Continuous-Time Algebraic Riccati Equation for Second-Order Systems

Abstract

This work presents closed-form methods to solve the Continuous-Time Algebraic Riccati Equation (CARE) for second-order systems. The standard CARE solution requires the computation of certain eigenvectors, which becomes expensive and erroneous as the size of the system increases. We mitigate these issues by developing closedform solutions to CARE expressed in terms of physically meaningful mass, damping, and stiffness matrices. We present two methods – one that requires the modal transformation of mass and stiffness matrices, and another that does not require this modal transformation. We show using hundreds of high-dimensional second-order systems that the proposed methods achieve similar or better accuracy compared to the state-of-the-art, while significantly reducing the computation time. We further substantiate our methods by illustrating their advantages when applied to engineering problems such as vibration control.