Balanced transformations: their computation and some applications

Balanced transformations, in which an input property of a linear system (e.g. controllability) is 'symmetrized' or 'balanced' with an output properly (e.g. observability) by making a suitable choice of basis, are considered. Some algorithms available for computing these transformations are examined for both the first-order and second-order forms of linear time-invariant systems. An important feature of these methods is the simultaneous diagonalization of the controllability/observability Gramians by solving two Lyapunov equations. Different application areas of balanced transformations are highlighted and some limitations and/or problems are pointed out.