Solution of the singular discrete regulator problem using eigenvector methods

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Abstract
The discrete regulator problem with a singular state matrix cannot be solved using either the eigenvectors or the sign function of the associated compound matrix, since then, the implicit equation defining the feedback matrix cannot be transformed into the regular discrete Riccati equation. In this paper, it is shown that the solution of a continuous Riccati equation, whose matrix parameters depend on the matrices of both discrete system and cost functional, is the solution of the implicit equation. Since the continuous Riccati equation has symmetric spectrum with respect to the imaginary axis, it has no singularity problem and therefore the matrix sign function method can be used for its solution.

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