Bounded real and positive real balanced truncation using Σ-normalised coprime factors

In this article, we will extend the method of balanced truncation using normalised right coprime factors of the system transfer matrix to balanced truncation with preservation of half line dissipativity. Special cases are preservation of positive realness and bounded realness. We consider a half line dissipative input-output system, with quadratic supply rate given by a nonsingular symmetric matrix Σ with the property that its positive signature is equal to the number of input components of the system. The transfer matrix of such system allows a Σ-normalised right coprime factorisation. We associate with such factorisation two Lyapunov equations, one of which is nonstandard, involving the indefinite matrix Σ. Balancing will be based on making the unique solutions of these two Lyapunov equations equal and diagonal. The diagonal elements will be called the Hankel Σ-singular values, because their squares are the nonzero eigenvalues of the composition of the `graph' Hankel operator, multiplication by Σ, and the adjoint graph Hankel operator. This method of balanced truncation will be shown to preserve stability, minimality, and half line dissipativeness. We will characterize the `classical' positive real and bounded real characteristic values in terms of the new Hankel Σ-singular values. Finally, we will derive one-step error bounds for the special case of balanced truncation of bounded real systems