A Low-rank ADI Algorithm for Solving Large-scale Non-symmetric Algebraic Riccati Equations
For researchers solving large-scale NAREs in control theory and engineering, this provides a unified and efficient algorithm that generalizes existing low-rank ADI methods.
This paper develops the first low-rank ADI algorithm for large-scale non-symmetric algebraic Riccati equations (NAREs) with low-rank solutions, achieving efficient and accurate solutions for problems up to order 10^6. The algorithm autonomously generates shifts and outperforms existing methods that only handle special cases.
This paper considers large-scale nonsymmetric continuous-time algebraic Riccati equations (NAREs) that admit low-rank solutions. Low-rank alternating direction implicit (ADI) methods have proven to be an efficient approach for solving several matrix equations, including Lyapunov equations, Sylvester equations, and symmetric Riccati equations. Although a low-rank algorithm for the Sylvester equation has been used as an inner loop in computing low-rank solutions of NAREs, no low-rank ADI algorithm currently exists for NAREs themselves. This paper fills this gap by developing a low-rank ADI algorithm for large-scale NAREs that admit a low-rank solution. Since Lyapunov equations, Sylvester equations, and symmetric Riccati equations are special cases of the NARE, the existing low-rank ADI methods in the literature are special cases of the more general low-rank ADI method proposed here. An automatic and computationally efficient method for shift generation is also discussed, and a subspace-accelerated projection approach is presented to generate shifts for subsequent iterations without user intervention. Once initialized with arbitrary shifts, the proposed algorithm solves large-scale NAREs autonomously, generating its own shifts. Numerical results are presented using benchmark example of order $10^6$, demonstrating the computational efficiency and accuracy of the proposed algorithm.