An Accelerated Residual ADI Method for Large-Scale Low-Rank Riccati Equations

This paper considers the acceleration of the residual alternating direction implicit (RADI) iteration for solving large-scale low-rank Riccati matrix equations arising from time-invariant control systems. A direct attempt to accelerate the ADI iteration by treating the feedback gain matrix as a fixed-point iterate typically leads to a relatively slow convergence of the norm of the residual matrix. To address this issue, we combine the feedback gain matrix with the residual matrix as the input of the RADI iteration and develop an accelerated RADI scheme based on this reformulation. The convergence of the accelerated RADI algorithm is established under a relatively mild assumption. Numerical experiments from engineering applications demonstrate that the proposed accelerated RADI algorithm with properly selected parameters is able to attain a prescribed residual level with fewer iterations and less computational time than the RADI method.