Model reduction of linear systems using extended balanced truncation

An extension to balanced truncation is presented. Balanced truncation is a standard method for model reduction and it has many good properties, such as preservation of model stability and a priori error bounds. Balanced truncation is done using controllability and observability Gramians. The Gramians can be found by solving a set of linear matrix inequalities. In this paper, we show that these linear matrix inequalities can be extended so that the number of decision variables are at least doubled. This leads to the concept of extended Gramians. It is shown that all the good properties of balanced truncation also hold for extended balanced truncation. It turns out that extended balanced truncation is especially useful when there are additional structure constraints in the model to be reduced. This is useful in frequency-weighted reduction, for example.