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

• Published in: IET control theory & applications Vol. 18; no. 7; pp. 834 - 845
• Main Authors: Zhang, Liangyin, Chen, Michael Z. Q., Gao, Zhiwei, Ma, Lifeng
• Format: Journal Article
• Language: English
• Published: Wiley 01-04-2024
• Subjects: combinatorial mathematics; linear quadratic control; matrix algebra; Riccati equations
• ISSN: 1751-8644; 1751-8652
• DOI: 10.1049/cth2.12618

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.
The explicit solutions of the algebraic Riccati equation for systems in controllable canonical form 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. If the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms.