A behavioral characterization of the positive real and bounded real characteristic values in balancing

In passivity preserving and bounded realness preserving model reduction by balanced truncation, an important role is played by the so-called positive real (PR) and bounded real (BR) characteristic values. Both for the positive real as well as the bounded real case, these values are defined in terms of the extremal solutions of the algebraic Riccati associated with the system, more precisely as the square roots of the eigenvalues of the product matrix obtained by multiplying the smallest solution with the inverse of the largest solution of the Riccati equation. In this paper we will establish a representation free characterization of these values in terms of the behavior of the system. We will consider positive realness and bounded realness as special cases of half line dissipativity of the behavior. We will then show that both for the PR and the BR case, the characteristic values coincide with the singular values of the linear operator that assigns to each past trajectory in the input–output behavior its unique maximal supply extracting future continuation. We will explain that the term ‘singular values’ should be interpreted here in a generalized sense, since in our setup the future behavior is only an indefinite inner product space.