Stability and error bounds for discrete time frequency weighted balanced truncation

This paper considers the frequency weighted model reduction of discrete time systems. It is first shown that the frequency weighted balanced truncation always produces a stable lower order model if the weighting function is either only in the input or only in the output. This is in sharp contrast with Enns´ continuous time results (1984) where certain controllability or observability condition of the reduced model is required to guarantee the stability of the reduced model. We then show that the frequency weighted balanced truncation can be used for relative or multiplicative error model reduction. Moreover, some explicit L_∞, norm error bounds are derived for the relative and multiplicative approximation errors in terms of the weighted singular values. Finally, it is shown that this model reduction method is in fact equivalent to the well known stochastic balance truncation