On the explicit Hermitian solutions of the continuous-time algebraic Riccati matrix equation for controllable systems

Abstract

This paper proposes explicit solutions for the algebraic Riccati matrix equation. For single-input systems in controllable canonical form, the explicit Hermitian solutions of the non-homogeneous Riccati equation are obtained using the entries of the system matrix, the closed-loop system matrix, and the weighting matrix. The unknown entries of the closed-loop system matrix are solved by scalar quadratic equations. For a homogeneous Riccati equation with a zero weighting matrix, the explicit solutions are proposed analytically in terms of the system eigenvalues. The advantages of the explicit solutions are threefold: first, if the system is controllable, the solution is directly given and the invariant subspaces of the Hamiltonian matrix are not required; second, if the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms; third, for a real system in the controllable canonical form, the non-negativity can be analysed for the explicit almost stabilizing solution.