Balanced representations and approximation of linear systems

The notion of a balanced state space is generalized for the class of L/sub 2/-trajectories of finite-dimensional, linear, time-invariant systems. Necessary and sufficient conditions are given for the existence of a state-space representation which is balanced with respect to general quadratic measures on the external behavior of systems in this class. In particular, balanced representations are derived for nonstable systems, and it is shown that traditional balanced representations are obtained as a special case.<>