Mixed H2/H∞ Balanced Truncations for Discrete Time Linear Systems

A mixed H2/H∞ balanced realization is proposed for discrete time systems, where an H2 Lyapunov and an H∞ Riccati equations have the same positive definite diagonal matrix. Based on this realization, a balanced truncation is possible, which gives a reduced-order model. It is shown that the H2 norm of any reduced-order model based on this realization is not greater than that of the original model. The H∞ norm of the reduced-order model is also bounded by the value which appears in the Riccati equation and gives an upper bound of the H∞ norm of the original model. A key idea for proving these properties is that the Riccati equation is rewritten by a matrix equation which does not contain inverse matrix via the bounded real lemma. Upper bounds of the H2 and the H∞ norms of the error system between the original model and the reduced-order model are also established. These properties are illustrated through a numerical example.