Continuous-Time Algebraic Riccati Equation Solution for Second-Order Systems

The continuous-time algebraic Riccati equation (ARE) is often utilized in control, estimation, and optimization. For a linear system with a second-order structure of size n, the ARE required to be solved to get the control values in standard control problems results in complex subequations in terms of the second-order system matrices. The computational costs of solving the algebraic Riccati equation through standard methods such as the Hamiltonian matrix pencil approach increase substantially as matrix sizes increase for a second-order system, due to the eigendecomposition of the $2n×2n$ system matrices involved. This work introduces a new solution that does not require the eigendecomposition of the $2n×2n$ system matrices, while satisfying all of the requirements of the solution to the Riccati equation (e.g., detectability, stabilizability, and positive semidefinite solution matrix).